Bevel Weight on edges
In the documentation we read:


  This edge property, a value between $0.0$ to $1.0$, is used by the Bevel Modifier to control the bevel intensity of the edges. This operator enters an interactive mode (a bit like transform tools), where by moving the mouse (or typing a value with the keyboard) you can set the bevel weight of selected edges. If two or more edges are selected, this operator alters the average weight of the edges.


See also bevel tutorial.